"Does anyone have evidence to show why the term 'blues' was selected when it became necessary to have a name for that kind of music?" The expression to have the blues meaning to be depressed was well known long before 1900. It became fairly popular for black folk singers to include the lyrics "I got the blues" in their songs only in roughly 1907 (in contrast to e.g. 1904 when we have no solid evidence of any doing so yet). People began talking about "blues" music to describe music like those particular songs (not e.g. all black folk music in general, or all sad folk music in general) in roughly 1909. Those particular songs happened to be similar to apparently earlier sad songs that didn't have the word "blues" in them, such as "Poor Boy Long Ways From Home" and "Joe Turner."

As far as we know the two main cliche early blues lyrics about having the "blues" may have been
(1) "I got the blues, too damn mean to cry" or close to that and
(2) "I got the blues, can't be satisfied" or close to that.

E.g.,
Howard Odum collected (1) and (2) from folk singers before 1909.
E.C. Perrow had (1) from 1909.
The very first copyrighted blues song was "The Blues (But I'm Too Blamed Mean To Cry)" by Smith and Brymn in early 1912.
"Baby Seals Blues" included (2) in 1912.
"I Wonder Where My Easy Rider's Gone" by Shelton Brooks included (1) in 1913.
W.P. Webb published in 1915 a folk song he had heard from the black singer Floyd Canada, including (1).
Newman Ivey White collected (1).
Langston Hughes, born in 1902, remembered hearing (2) as "a kid."
Lovie Austin, born in 1887, used (2) in "Chicago Bound Blues."
Early-born artists who sang "can't be satisfied" include Mamie Smith ("Don't Care Blues"), John Hurt, Robert Wilkins, Miles Pratcher, Sylvester Weaver, Barbecue Bob, William Harris, Sam Butler, Lottie Kimbrough, Willie Newbern, Alex Moore, Bill Broonzy.
Ivy Smith recorded "Too Mean To Cry Blues" in 1927.
Curly Weaver sang (1) in 1928.
Trixie Smith sang "I'm too darn mean to cry" in "Freight Train Blues."

"blues has always included a lot of happy dance music" As far as I know all the earliest blues songs (what I'd call blues songs) had sad lyrics. I'm talking about well before Lloyd Garrett wrote lyrics to "Dallas Blues" in 1918. An example of what I consider an early blues lyric is this published by Howard Odum in 1911:

"Got up in the morning, couldn't keep from crying
Got up in the morning, couldn't keep from crying
Got up in the morning, couldn't keep from crying
Thinking about that brown-skin man of mine....
So she laid in jail back to the wall
So she laid in jail back to the wall
So she laid in jail back to the wall
This brown-skin man cause of it all...."

People talked of "blue notes" in the 1910s in connection with what was already known as blues music.